Multiple choice

The equation of the circle and its chord are respectively${ x }^{ 2 }+{ y }^{ 2 }={ a }^{ 2 }$ are $x cos\alpha +y sin\alpha =p$. The equation of the circle of which this chord is diameter is-

  1. ${x}^{2}+{y}^{2}-{ 2 }_{ PX }cos\alpha -{ 2 }_{ py }sin\alpha +{ 2 }_{ { p }^{ 2 } }-{ a }^{ 2 }=0$
  2. ${x}^{2}+{y}^{2}-{ 2 }_{ PX }cos\alpha -{ 2 }_{ py }sin\alpha + { p }^{ 2 } -{ a }^{ 2 }=0$
  3. ${x}^{2}+{y}^{2}-{ 2 }+{ PX }cos\alpha -{ 2 }+{ py }sin\alpha +{ 2 }_{ { p }^{ 2 } }-{ a }^{ 2 }=0$

  4. None of these
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B Correct answer
AI explanation

The equation of any circle passing through the intersection of the given circle x^2 + y^2 = a^2 and the chord x cos(alpha) + y sin(alpha) = p can be written as a family of curves: x^2 + y^2 - a^2 + k(x cos(alpha) + y sin(alpha) - p) = 0. To find the specific value of k for which the chord acts as a diameter, we need the center of this new circle to lie on the chord itself. The center of the family of curves is located at (-k*cos(alpha)/2, -k*sin(alpha)/2); substituting this point into the chord equation gives (-k/2)(cos^2(alpha) + sin^2(alpha)) = p, which simplifies to -k/2 = p and means k = -2p. Substituting k = -2p back into the family equation yields x^2 + y^2 - 2px cos(alpha) - 2py sin(alpha) + p^2 - a^2 = 0.

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